Circular Motion#

If \(P\) is observed in \(\mathbf{e}\) to move in a circle at a constant rate in the \(\mathbf{e}_1,\mathbf{e}_2\) plane then \(P\) has the representation \(\left(t, (\sin{(\omega t)},\cos{(\omega t)},0)\right)\). Then in the \(\mathbf{e}'\) frame that is moving in the \(\mathbf{e}_3\) direction at a constant rate \(v\) we see that this point has the representation \(\left(t, (\sin{(\omega t)},\cos{(\omega t)},-vt)\right)\) and hence is observed to be moving in a helical path.

Import following libraries on your notebook

[4]:

import plotly.graph_objects as go
import numpy as np

Since we have explained the syntax and functions going to use here I am leaving the code only. Follow the previous tutorials for better understanding. Hit the play button and see!

[6]:

# Define constants
v = np.array([0, 0, 1])  # Constant velocity vector
w = 4  # Angular frequency

# Generate time values from 0 to 5 with 101 points
tt = np.linspace(0, 5, 101)

# Initialize an array to store particle positions
zeta = np.zeros((len(tt), 3))

# Calculate particle positions based on the given formula
for i, t in enumerate(tt):
    # Calculate acceleration vector based on sinusoidal motion
    a = np.array([np.sin(w * t), np.cos(w * t), 0])

    # Update particle position using the formula (a - v * t)
    zeta[i, :] = (a - v * t)

# Create a particle animation using the calculated positions
fig = create_particle_animation(zeta)

# Display the animation
fig.show()